Inset while keeping part of the shape intact
One simple way is to temporarily extrude the shape and inset part of extrusion which will create edges on the boundary. So instead of a plane:


You use the same form but extruded (or just extrude one edge where you need edges to touch the boundary)


Then select top faces plus one on the side and run Inset:


Dissolve unnecessary edges on the side as required (or delete bottom vertices to get rid of extrusion), note that dissolving might create Ngons which are better to avoid.